Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-phase Darcy Flows in Fractured Porous Media
Résumé
This work extends the Vertex Approximate Gradient (VAG) discretization of multi-phase Darcy flow models in order to take into account discrete fracture networks (DFN). We consider the asymptotic model for which the fractures are represented as interfaces of codimension one immersed in the matrix domain with continuous pressures at the matrix fracture interface. Our discretization takes into account general polyhedral meshes, general discrete fracture networks, the anisotropy of the matrix and of the fracture permeability fields, and discontinuous rocktypes. Compared with Control Volume Finite Element (CVFE) approaches, the VAG scheme has the advantage to avoid the mixing of the fracture and matrix rocktypes at the interfaces between the matrix and the fractures, while keeping the low cost of a nodal discretization on unstructured meshes. The convergence of the scheme to a weak solution of the model is proved for arbitrary choices of the volumes at the nodal unknowns assuming the non degeneracy of the relative permeabilities and a network of planar fractures. Numerical experiments exhibiting the efficiency of the VAG discretization are presented for 3D two phase flow simulations in fractured networks with high permeability contrasts between the matrix and the fractures including an application to tight gas recovery.